Bayesian Adaptive Methods

1. Bayesian Adaptive Methods Carrie Deis Nadine Dewdney

2. Overview • Phase I clinical trials • Standard Designs • Adaptive Designs • Bayesian Approach • Traditional vs. Bayesian • Hybridization • FDA Guidance • Conclusion

3. Phase I • Conducted to determine toxicity for the dosing of the new intervention • First time the drug is tested in humans • Small number of patients, 20 to 50 • Depending on the nature of the new drug, patients are usually healthy volunteers • A higher dose is assumed to be more effective • Goal is to find maximum tolerable dose (MTD)

4. Phase I • Known prior to the start of the trial: • Starting dose • Toxicity profile and Dose-limiting toxicity • Target toxicity level • Dose Escalation Scheme • Starting dose commonly chosen as: • 1/3 lowest toxic dose in dogs • 1/10 of the LD10 in mice • Dose escalation is done incrementally • Increments are pre-determined • Modified Fibonacci sequence – increase rate diminishing as the dose gets higher

5. Standard Design • Patients are assigned to dose levels according to predefined rules • Allow for only escalation and de-escalation of dose • Doses selected such that, D1,…, DK would be close to MTD • MTD is determined statistically as the dose at which 1/3 of the subjects develop toxicity

6. Standard Design • Subjects are randomized • The number of subjects, ri, developing toxicity would be observed • pi = ri/ni, is used to calculate the proportions exhibiting toxicity • Dose-response is modeled based on the probability of toxicity • The MTD would be fitted to this model

7. Standard Design • Ethical concerns with the traditional approach • Patients might be treated excessively and unnecessarily at low doses • Too many patients may be treated at doses that are too high or too low • Highly likely most subjects are treated at low doses • Not clear that the estimated MTD is the correct dose

8. Adaptive Designs • Adjustments and modifications can be made after the trial has started • Does not affect the integrity of the trial • Goal is to improve upon the probability of success of the trial and correctly identify the clinical benefits of the intervention under investigation • Prospective adaptations include • Stopping a trial early for safety or lack of efficacy • Dropping the loser - Inferior treatments dropped • Sample size re-estimation

9. Adaptive Designs • Modifications hypothesis might be necessary • Inclusion/exclusion criteria • Dose/regimen • Treatment duration • Endpoints • Several types of adaptive designs • Group sequential • Sample size adjustable design • Drop-the-losers design • Adaptive treatment allocation design • Bayesian adaptive methods

10. Bayesian Approach • Based on Bayes Theorem:  • Expresses how a subjective degree of belief should rationally change to account for evidence • Used as a statistical inferential tool in adaptive designs • Strength of the Bayesian approach • Decision on trial continuation is made as data accumulates • Sample size not determined in advance although a maximum size might be specified • Drawbacks • Analysis after each subject is treated

11. Bayesian Approach • Calculates the predictive probability that the patient will respond to treatment • Specifies a prior distribution then updates it as information becomes available • Uses the likelihood function and the prior distribution to obtain a posterior distribution • MTD is determined from the posterior distribution • Studies are based on costs and public health benefits

12. Bayesian Approach • Prior Distribution • Logistic model: p(d) = exp (3 + ad)/[ 1 + exp(3 + ad)] • Power model: p(d) = dexp(a) • p(d) is the probability of DLT • d is the dose • a is a model parameter

13. Bayesian Approach

14. Bayesian Approach • Once the posterior distribution is calculated: • The MTD is revised based on the distribution of a • The mode of the posterior distribution is used to estimate the next dose • Each patient is treated at the dose which is closest to the MTD • Toxicity profile is updated after each patient is treated • The sequence is repeated until a precise estimate of parameter a is obtained or the sample size is exhausted

15. Traditional vs. Bayesian • Example: A dose-finding escalation design from an oncology trial • Traditional approach • The 3+3 traditional escalation rule (TER) • Bayesian approach • The continual reassessment method (CRM) • The objective is to determine the MTD for a new drug using the least amount of patients

16. Traditional vs. Bayesian • Results from animal studies: • The dose limiting toxicity rate was determined to be 1% for the starting dose of 25 mg/m2, 1/10 of the lethal dose • The MTD is estimated to be 150 mg/m2 • The dose limiting toxicity rate is defined as 0.25 • Selected Model: A logistic toxicity model • Dose sequence was chosen with interim factors = 2, 1.67, 1.33, 1.33, 1.33, 1.33, 1.33, 1.33, 1.33

17. Traditional vs. Bayesian Summary of simulation results for the designs

18. Traditional vs. Bayesian • Both approaches underestimate the true MTD • However, the Bayesian approach was much closer to the true value for all dose levels • At all three dose levels the Bayesian approach required less patients • The mean number of DLTs for the Bayesian approach was either less than or equal to the traditional approach at all dose levels • The Bayesian CRM approach proved to be more favorable

19. Hybridization • The Bayesian approach can be used alone or as a hybrid with the classic approach • As a hybrid, the Bayesian approach is used to increase the probability of success • Example: Two-arm parallel design • Compares a test treatment and a control • Use data from 3 clinical trials with similar sample sizes • Prior probabilities for the effect size are 0.1, 0.25, and 0.4 with 1/3 probability for each trial

20. Hybridization • The classic approach: • Mean of the effect size, = 0.25, is used to calculate the sample size. For the design with β = 0.2: • The Bayesian approach: • The power of the effect size is Φ is the c.d.f. of the standard normal distribution • Prior, π(ε), is the uncertainty of ε, the expected power

21. Hybridization + • Assuming, one-sided α = 0.025, • Pexp =0.66 • With the hybrid approach the power is less than the 80% power stated in the frequentist approach, recall β = 0.2. In order to reach the expected power of 80%, the sample size needs to be increased • The Bayesian approach piece is used to increase the probability of success given that the final criterion is p ≤ α = 0.025

22. FDA Guidance – Medical Devices • Prior information and Assumptions • Criterion for success for safety and effectiveness • Justification for the proposed sample size • Prior probability of the study claim • This is the probability of the study claim before seeing any new data, and it should not be too high • Ensures the prior information does not overwhelm the current data, potentially creating a situation where unfavorable results from the proposed study get masked by a favorable prior distribution • Program Code

23. FDA Guidance – Medical Devices • Operating characteristics • Provide tables of the probability of satisfying the study claim, given “true” parameter values and sample sizes for the new trial • Provides an estimate of the probability of a type I error in the case where the true parameter values are consistent with the null hypothesis, or power in the case where the true parameter values are consistent with the alternative • Effective Sample Size • Quantifies the efficiency you are gaining from using the prior information and gauges if the prior is too informative

24. Conclusion • Bayesian full approach is more beneficial in Phase I studies • Inherent adaptive nature of the design • Conditions are more dynamic than other phases and the flexible nature of the Bayesian approach allows for unexpected changes • Produces a posterior probability which is useful in decision making and the transitioning from one phase to the next • Dose levels can be modified which could be beneficial for a phase I cancer study

25. Conclusion • Even without using a full Bayesian method, hybridization results in increased probability of success in trials • Maintaining the validity and integrity of the study and control of the type I error in applications of the method is important • Feasibility should be evaluated in order to prevent abuse of this method in applications such as endpoints or hypotheses changes • The FDA is cautious of the growing trend of Bayesian designs and continues to set guidelines for its use in Phase I trials

26. References • Chang, Mark (2008). Adaptive Design Theory and Implementation Using SAS and R. Boca Raton: Chapman & Hall/CRC • Berry, Scott M., Carlin, Bradley P., Lee, J.Jack, Muller, Peter (2011). Bayesian Adaptive Methods for Clinical Trials. Boca Raton: Chapman & Hall/CRC • Chow, Shein-Chung and Chang, Mark (2008). Adaptive Design Methods in Clinical Trials – A Review. Orphanet Journal of Rare Diseases, 3 11 • Cook, Thomas D. and DeMets, David L. (2008). Introduction to Statistical Methods for Clinical Trials. Boca Raton: Chapman & Hall/CRC • The FDA Center for Drug Evaluation and Research, and Center for Biologics Evaluation and Research, Guidance for Industry: Adaptive Design Clinical Trials for Drugs and Biologics: www.fda.gov

27. Questions